Remark 1 
The iteration complexity can be improved to \(O(\epsilon ^{-2})\) , whenever the original problem (REF ) has the same form as problem (REF ); see also [1]}, [2]}, [3]} for related discussions.
Specifically, we can avoid introducing variables \(z\)  with penalty weights \(O(\epsilon ^{-2})\) , whenever formulation (REF ) contains a variable block, say \(x_T\) , that is constrained only via the linear coupling constraints; i.e., \(X_T = \mathbb {R}^{n_T}\) , and its coupling matrix satisfies \(A_T = \mathrm {I}\)  (or more generally, its image \(\mathop {\mathrm {Im}}\left(A_T\right) \supseteq \mathop {\mathrm {Im}}\left([b, A_1, \ldots , A_{T-1}]\right)\) ), and its objective satisfies \(\nabla ^2 f_T (x_T) \preceq M\mathrm {I}\)  for all \(x \in X\) , for some fixed \(M > 0\) .

